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A survey of numerical cubature over triangles

Description: This survey collects together theoretical results in the area of numerical cubature over triangles and is a vehicle for a current bibliography. We treat first the theory relating to regular integrands and then the corresponding theory for singular integrands with emphasis on the ``full comer singularity.`` Within these two sections we treat successively approaches based on transforming the triangle into a square, formulas based on polynomial moment fitting, and extrapolation techniques. Within each category we quote key theoretical results without proof, and relate other results and references to these. Nearly all the references we have found may be readily placed in one of these categories. This survey is theoretical in character and does not include recent work in adaptive and automatic integration.
Date: December 31, 1993
Creator: Lyness, J. N. & Cools, R.
Partner: UNT Libraries Government Documents Department

On cylindrically converging shock waves shaped by obstacles

Description: Motivated by recent experiments, numerical simulations were performed of cylindrically converging shock waves. The converging shocks impinged upon a set of zero to sixteen regularly space obstacles. For more than two obstacles the resulting diffracted shock fronts formed polygonal shaped patterns near the point of focus. The maximum pressure and temperature as a function of number of obstacles were studied. The self-similar behavior of cylindrical, triangular and square-shaped shocks were also investigated.
Date: July 16, 2007
Creator: Eliasson, V; Henshaw, W D & Appelo, D
Partner: UNT Libraries Government Documents Department

Multigrid for refined triangle meshes

Description: A two-level preconditioning method for the solution of (locally) refined finite element schemes using triangle meshes is introduced. In the isotropic SPD case, it is shown that the condition number of the preconditioned stiffness matrix is bounded uniformly for all sufficiently regular triangulations. This is also verified numerically for an isotropic diffusion problem with highly discontinuous coefficients.
Date: February 1, 1997
Creator: Shapira, Yair
Partner: UNT Libraries Government Documents Department

INTEGRATING A LINEAR INTERPOLATION FUNCTION ACROSS TRIANGULAR CELL BOUNDARIES

Description: Computational models of particle dynamics often exchange solution data with discretized continuum-fields using interpolation functions. These particle methods require a series expansion of the interpolation function for two purposes: numerical analysis used to establish the model's consistency and accuracy, and logical-coordinate evaluation used to locate particles within a grid. This report presents discrete-expansions for a linear interpolation function commonly used within triangular cell geometries. Discrete-expansions, unlike a Taylor's series, account for interpolation discontinuities across cell boundaries and, therefore, are valid throughout a discretized domain. Verification of linear discrete-expansions is demonstrated on a simple test problem.
Date: April 1, 2000
Creator: WISEMAN, J. R. & BROCK, J. S.
Partner: UNT Libraries Government Documents Department

Cardinality Bounds for Triangulations With Bounded Minimum Angle

Description: We consider bounding the cardinality of an arbitrary triangulation with smallest angle {alpha}. We show that if the local feature size (i.e. distance between disjoint vertices or edges) of the triangulation is within a constant factor of the local feature size of the input, then N < O(1/{alpha})M, where N is the cardinality of the triangulation and M is the cardinality of any other triangulation with smallest angle at least {alpha}. Previous results had an O(1/{alpha}{sup 1/{alpha}}) dependence. Our O(1/{alpha}) dependence is tight for input with a large length to height ratio, in which triangles may be oriented along the long dimension.
Date: May 1, 1994
Creator: Mitchell, S. A.
Partner: UNT Libraries Government Documents Department

Linear-size nonobtuse triangulation of polygons

Description: We give an algorithm for triangulating n-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than {pi}/2. The number of triangles in the triangulation is only 0(n), improving a previous bound of 0(n{sup 2}), and the worst-case running time is 0(n log{sup 2} n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm.
Date: May 1, 1994
Creator: Bern, M.; Mitchell, S. & Ruppert, J.
Partner: UNT Libraries Government Documents Department

Holliday Triangle Hunter (HolT Hunter): Efficient Software for Identifying Low Strain DNA Triangular Configurations

Description: Synthetic DNA nanostructures are typically held together primarily by Holliday junctions. One of the most basic types of structures possible to assemble with only DNA and Holliday junctions is the triangle. To date, however, only equilateral triangles have been assembled in this manner - primarily because it is difficult to figure out what configurations of Holliday triangles have low strain. Early attempts at identifying such configurations relied upon calculations that followed the strained helical paths of DNA. Those methods, however, were computationally expensive, and failed to find many of the possible solutions. I have developed a new approach to identifying Holliday triangles that is computationally faster, and finds well over 95% of the possible solutions. The new approach is based on splitting the problem into two parts. The first part involves figuring out all the different ways that three featureless rods of the appropriate length and diameter can weave over and under one another to form a triangle. The second part of the computation entails seeing whether double helical DNA backbones can fit into the shape dictated by the rods in such a manner that the strands can cross over from one domain to the other at the appropriate spots. Structures with low strain (that is, good fit between the rods and the helices) on all three edges are recorded as promising for assembly.
Date: April 16, 2012
Creator: Sherman, W.B.
Partner: UNT Libraries Government Documents Department

The RQM triangle: A paradigm for relativistic quantum mechanics

Description: A simple way to relate Lorentz transformations to a finite and discrete model of counter firings is presented. We first abstract from counter firings with finite resolution to rational fraction velocities and a finite step-length. Discrete Lorentz transformations and quantized rotations follow. These are encoded by three integers and a triangle with integer sides: the RQM Triangle. The double slit experiment allows us to observe quantized space steps and measure invariant step lengths for any particle in rational ratio to one such length for any convenient reference particle. A discrete version of the Mandelstam analysis of elastic and anelastic scattering is implied. The results fit naturally into the bit-string formalism used in earlier work.
Date: February 1, 1992
Creator: Noyes, H. P.
Partner: UNT Libraries Government Documents Department

Refining a triangulation of a planar straight-line graph to eliminate large angles

Description: Triangulations without large angles have a number of applications in numerical analysis and computer graphics. In particular, the convergence of a finite element calculation depends on the largest angle of the triangulation. Also, the running time of a finite element calculation is dependent on the triangulation size, so having a triangulation with few Steiner points is also important. Bern, Dobkin and Eppstein pose as an open problem the existence of an algorithm to triangulate a planar straight-line graph (PSLG) without large angles using a polynomial number of Steiner points. We solve this problem by showing that any PSLG with {upsilon} vertices can be triangulated with no angle larger than 7{pi}/8 by adding O({upsilon}{sup 2}log {upsilon}) Steiner points in O({upsilon}{sup 2} log{sup 2} {upsilon}) time. We first triangulate the PSLG with an arbitrary constrained triangulation and then refine that triangulation by adding additional vertices and edges. Some PSLGs require {Omega}({upsilon}{sup 2}) Steiner points in any triangulation achieving any largest angle bound less than {pi}. Hence the number of Steiner points added by our algorithm is within a log {upsilon} factor of worst case optimal. We note that our refinement algorithm works on arbitrary triangulations: Given any triangulation, we show how to refine it so that no angle is larger than 7{pi}/8. Our construction adds O(nm+nplog m) vertices and runs in time O(nm+nplog m) log(m+ p)), where n is the number of edges, m is one plus the number of obtuse angles, and p is one plus the number of holes and interior vertices in the original triangulation. A previously considered problem is refining a constrained triangulation of a simple polygon, where p = 1. For this problem we add O({upsilon}{sup 2}) Steiner points, which is within a constant factor of worst case optimal.
Date: May 13, 1993
Creator: Mitchell, S. A.
Partner: UNT Libraries Government Documents Department

Passive localization processing for tactical unattended ground sensors

Description: This report summarizes our preliminary results of a development effort to assess the potential capability of a system of unattended ground sensors to detect, classify, and localize underground sources. This report also discusses the pertinent signal processing methodologies, demonstrates the approach with computer simulations, and validates the simulations with experimental data. Specific localization methods discussed include triangulation and measurement of time difference of arrival from multiple sensor arrays.
Date: September 1995
Creator: Ng, L. C. & Breitfeller, E. F.
Partner: UNT Libraries Government Documents Department

The RQM triangle: A paradigm for relativistic quantum mechanics

Description: A simple way to relate Lorentz transformations to a finite and discrete model of counter firings is presented. We first abstract from counter firings with finite resolution to rational fraction velocities and a finite step-length. Discrete Lorentz transformations and quantized rotations follow. These are encoded by three integers and a triangle with integer sides: the RQM Triangle. The double slit experiment allows us to observe quantized space steps and measure invariant step lengths for any particle in rational ratio to one such length for any convenient reference particle. A discrete version of the Mandelstam analysis of elastic and anelastic scattering is implied. The results fit naturally into the bit-string formalism used in earlier work.
Date: February 1, 1992
Creator: Noyes, H.P.
Partner: UNT Libraries Government Documents Department

A transient, quadratic nodal method for triangular-Z geometry

Description: Many systematically-derived nodal methods have been developed for Cartesian geometry due to the extensive interest in Light Water Reactors. These methods typically model the transverse-integrated flux as either an analytic or low order polynomial function of position within the node. Recently, quadratic nodal methods have been developed for R-Z and hexagonal geometry. A static and transient quadratic nodal method is developed for triangular-Z geometry. This development is particularly challenging because the quadratic expansion in each node must be performed between the node faces and the triangular points. As a consequence, in the 2-D plane, the flux and current at the points of the triangles must be treated. Quadratic nodal equations are solved using a non-linear iteration scheme, which utilizes the corrected, mesh-centered finite difference equations, and forces these equations to match the quadratic equations by computing discontinuity factors during the solution. Transient nodal equations are solved using the improved quasi-static method, which has been shown to be a very efficient solution method for transient problems. Several static problems are used to compare the quadratic nodal method to the Coarse Mesh Finite Difference (CMFD) method. The quadratic method is shown to give more accurate node-averaged fluxes. However, it appears that the method has difficulty predicting node leakages near reactor boundaries and severe material interfaces. The consequence is that the eigenvalue may be poorly predicted for certain reactor configurations. The transient methods are tested using a simple analytic test problem, a heterogeneous heavy water reactor benchmark problem, and three thermal hydraulic test problems. Results indicate that the transient methods have been implemented correctly.
Date: June 1, 1993
Creator: DeLorey, T. F.
Partner: UNT Libraries Government Documents Department